Calculus

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1. Set Theory and Functions

1.1 Generalized Distributive and De Morgan’s Laws

Consider a family of sets , with , where is an index set.

Distributive Law:

De Morgan’s Laws:

Proof of the Distributive Law:

Use (1) as example:

Let's prove the first equation.

Proof of De Morgan’s Law:

Use (3) as example:

Let's prove the third equation.

1.2 Fundamental Properties

1.2.1 Spare Rational Number

Consider the set of rational numbers on the interval

Let’s take a value . The points are in intervals with an interval length of . The total length of the union of these intervals is

on leaves at least a length of empty space. We can take to be arbitrarily small. So, the real line is filled with irrational numbers everywhere.

1.2.2 Upper and Lower Bounds of

Since is always false, then the proposition is always true. This means the upper bound of is any real number. By convention:

1.2.3 Countability

Definition: A set is countable if its elements can be arranged into a sequence according to a certain rule.

Every finite set is countable, but not every infinite set is countable.

Consider , then is countable. (Diagonalization Principle) We can arrange them in a matrix-like form: (Remove duplicate elements.)

1.2.4 Cartesian Product

Definition: For any element in set and any element in set , a corresponding ordered pair is formed, and the set of all such pairs is called the Cartesian product of and , denoted as .

1.2.5 Functions

A function is defined by its rule: .

The conditions for a function are:

  1. Domain

  2. Codomain

  3. Uniqueness

Injective (One-to-one): No two values map to one.

Surjective (Onto): Every element is mapped.

Bijective (One-to-one correspondence): Both injective and surjective.

Proposition for injectivity: .

Note: Each element in the domain must map to a unique image, but a unique image does not require a unique pre-image.

Boundedness: A function is bounded if there exists such that for all , . is an upper bound of . is a lower bound of . is an upper bound. is a lower bound.

Monotonicity: Strictly monotonic Monotonic.

Periodicity, Parity, …

2. Sequence

2.1 Continuity of Real Number

Natural numbers are defined by the following 5 axioms (Peano Axioms):

  • 1 is a natural number.

  • Every natural number has a successor.

  • 1 is not the successor of any natural number.

  • The successor is unique.

  • (Principle of Mathematical Induction) If is a proposition about natural numbers such that

Natural numbers (closed to plus and multiply) are extended to integers () by subtraction, and extended to rational numbers () by division.

2.1.1 Upper Bounds and Lower Bounds

Definition: Let be a non-empty set. If s.t. is an upper bound of . If s.t. is a lower bound of .

Upper bound: Least upper bound sup .
Lower bound: Greatest lower bound inf .

Properties of the least upper bound (sup): (Let )

  • is an upper bound of . .
  • Any number smaller than is not an upper bound. .

2.1.2 Supre- /Infimum Existence Theorem

Theorem: Every non-empty set of real numbers that is bounded above has a least upper bound (supremum). Every non-empty set of real numbers that is bounded below has a greatest lower bound (infimum).

Pf.For all ,

Here, denotes the integer part, and denotes the fractional part.

Let

Then any set of real numbers can be represented by definite infinite decimals:

Assume is bounded above. Let be the largest integer among the integer parts of the elements in . Then define: Let be the largest first decimal digit among the elements in . Then define:

…

This implies:

Let . We now prove that is the supremum of .

Step 1: Prove is an upper bound of .

If , then exactly one of the following two cases holds:

  • (*) There exists such that
  • (**) For all ,

If case (*) holds, then .

If case (**) holds, then by comparing digit by digit, we get .

Step 2: Prove that for every , there exists such that .

For any , there exists such that .

Take . Then:

Therefore:

That is, .

Theorem: The supremum and infimum of a non-empty bounded set of real numbers are unique.

2.2 Sequence Limits and Infinitesimals

2.2.1 Definition

Let be a given sequence, . If , such that , . Denoted as

2.2.2 Properties

  • Uniqueness: The limit of a convergent sequence must be unique

Pf.: Take

When , we have and , which is a contradiction.
  • Boundedness: A convergent sequence must be bounded

Pf.: Assume converges. Take , such that , , then

Take Clearly (discussion for all terms)

Note: A bounded sequence may not necessarily converge.

  • Order Preservation: If , both converge, , , and , then ,

Pf.:

Take , by Then Similarly, by Then we have

Def: Sequence with limit equals to 0 is called a infinitesimal.

2.3 Arthematic Operations of Limit

2.3.1 Pre-conditions

.

These introduce two relations:

We can also say and are infinitesimals, for their limit is 0.

2.3.2 Summation

Pf.

2.3.3 Production

Pf.

2.3.4 Division

Pf.

Take such that , . That is, Thus , . Then Let That is,

2.3.5 Squeeze Theorem

If satisfy for , and , then .

Pf.:

Let , then for , , . Therefore That is, ,

2.4 Infinite Sequences

2.4.1 Definition

A sequence is called an infinite sequence if , such that , If , it is called a positive infinite sequence: If , it is called a negative infinite sequence:

Note: Infinite sequence ≠ Unbounded sequence. Counterexample: 1,1,2,1,3,1,4,1,…

2.4.2

Theorem: If , then is an infinite sequence if and only if is an infinitesimal sequence.

Pf.

(i) If is infinite is infinitesimal

That is, , hence:

(ii) If is infinitesimal is infinite

That is, , hence is infinite.

2.4.3

Theorem: If is infinite and for , (uniformly positive lower bound), then is also infinite.

Pf.

Let . Then for , we have

Corollary: If is infinite and , then is infinite.

2.4.4 Arithmetic Operations of Infinite Sequences

2.4.5 Stolz Theorem

Definition: If satisfies , , we call monotonically increasing.

If , it is strictly increasing.

Stolz Theorem: Let be a strictly increasing positive infinite sequence, and

Then

Pf.Case 1:

Let , When : Similarly: Thus: Since is positive infinite: Let , then : That is,

Case 2:

Let , then : Since is infinite: Let , then :

Case 3: (Proof omitted, similar to Case 2)

Example 1: If , then

Example 2:

2.5 Convergence Criteria (Real Number Continuity Theorem)

2.5.1 Monotone Convergence Theorem

A monotone bounded sequence must converge.

Pf.Assume is monotonically increasing and bounded above, hence has a supremum.

Let , we prove . By definition of supremum: For , and , hence ,

Example: Nested Radical Limit

Prove the existence of (n nested radicals)

Let , then ,

(i) Boundedness: Prove ,

, is obvious. Assume , then , holds.

(ii) Monotonicity: Prove

Assume Therefore is monotone bounded and convergent

2.5.2 Fibonacci Sequence Growth Rate

Fibonacci sequence: , . Let , find If , then ; if , then . Since Thus By Monotone Convergence Theorem, and exist Similarly, Therefore

2.5.3 The Number **

Pf.: is increasing, is decreasing, and both have the same limit:

Therefore both sequences are monotone bounded

2.5.4 p-Series Convergence

Let , , discuss convergence of

Case 1:

Let , Since , converges

Case 2:

2.5.5 Harmonic Series and Euler’s Constant**

Let , prove converges. Known: , . Taking logarithms: This indicates Hence monotonicly decays. Let (Euler's constant)

2.5.6 Nested Interval Theorem

Definition: A sequence of closed intervals satisfying two conditions:

Nested Interval Theorem: If is a nested sequence of closed intervals, then there exists a unique belonging to every , and

Pf.By the definition of nested intervals,

That is, and is bounded above by ; and is bounded below by Therefore both and have limits Let By the Monotone Convergence Theorem and uniqueness of limits, Hence If there exists another satisfying Then by the Squeeze Theorem,

Example: Prove is uncountable.

Method 1: Nested Interval Theorem

Proof by contradiction. Assume is countable Let Take such that Trisect into three intervals: (Trisection ensures does not belong to at least one interval) Choose one interval that doesn't contain , call it Repeat this process to obtain This gives us where:
  • This is a nested sequence of closed intervals
  • ,
By the Nested Interval Theorem, such that , Therefore , , i.e., but , contradiction

Method 2: Diagonal Argument

It suffices to prove is uncountable Proof by contradiction. Assume it's countable Let the elements be: Construct where , Then , Therefore is uncountable

2.7 Subsequence

2.7.1 Definition

Suppose is a sequence, and is a strictly increasing sequence of positive integers, then is called a subsequence of , denoted as . Obviously, we know if , then

Theorem: If converges to , then any subsequence of also converges to .

Pf., there exists , such that . Since , thus . Therefore, .

Corollary: If has two subsequences converging to different limits, then diverges. (Contrapositive of the above theorem)

Theorem: converges to if and only if every subsequence of has a subsequence that converges to .

Pf.

Sufficiency is obvious.

Necessity: By contradiction, assume that does not converge to .

That is, , such that , there exists with .

, such that . , such that . (taking as ) , such that . , , .

Any subsequence of does not converge to .

2.7.2 Bolzano-Weierstrass Theorem.

A bounded sequence must have a convergent subsequence.

Pf.
Suppose is bounded, i.e., , such that , . Bisect the interval into: , . At least one of them contains infinitely many , denote it as .

Repeat this process to obtain a nested sequence of closed intervals , where each contains infinitely many .

Take .

Take , with .

Take , with .

…

We obtain a subsequence satisfying . By the Nested Interval Theorem, , such that . By the Squeeze Theorem, .

2.7.3

Theorem. If is unbounded, then there exists a subsequence , such that .

Take , such that .

Take , such that .

…

We obtain a subsequence .

is an infinite sequence.

2.7.4 Theorem

If is an infinite sequence, then there exists a subsequence that is also an infinite sequence.

Take , such that

Take , such that

…

We obtain a subsequence .

is an infinite sequence.

2.8 Cauchy Convergence Principle

2.8.1 Cauchy Sequence

Definition (Cauchy Sequence):

Example:

Pf.

So

Take , then

Example:

Pf.

If we take , then no such exists

2.8.2 Cauchy Convergence Principle:

Cauchy Convergence Principle:

Pf.

  • Sufficiency:

Assume
Then

For any , we have

Hence, is a Cauchy sequence.

  • Necessity:
    First, prove that any Cauchy sequence is bounded. By the definition of a Cauchy sequence, Then Let Then

By the Bolzano-Weierstrass theorem, there exists a subsequence such that . Now, , by the Cauchy sequence definition,

Also,

For any , take sufficiently large such that . Then

Thus, .

Contraction Condition: If such that , then converges.

Pf. Show that is a Cauchy sequence. Assume without loss of generality that . Then:

3. Functional Limit

3.1 Definition and Properties

3.1.1 Definition of functional limit**

Definition: Suppose is defined in a certain deleted neighborhood of , i.e., , If , we have . Then is called the limit of at . Denoted as Note: is possible.

e.g. Prove

Pf.

3.1.2 Heine Theorem**

The necessary and sufficient condition for is that: for any sequence satisfying and , the corresponding sequence of functional values satisfies Pf. Necessity first. Assume , take any satisfying and . By the definition of functional limit, By the definition of sequential limit, Therefore , i.e., . Sufficiency. Proof by contradiction. Assume , i.e., , s.t. Take , s.t. and . Take , s.t. and . Obtain satisfying , but and
Theorem: The necessary and sufficient condition for the existence of is that for any sequence satisfying and , all converge.

3.2 Properties of Functional Limit

  • Uniqueness: If are both limits of at , then .
Pf. Take any and Then and . By the uniqueness of sequential limits, .
  • Squeeze Theorem: If , is satisfied. And , then .
Pf. Take any satisfying . Therefore . By Heine Theorem, . By the sequential Squeeze Theorem, . By Heine Theorem, .

  • Four Arithmetic Operations: Suppose .

Pf. Take any satisfying and . By Heine Theorem, . By the four arithmetic operations of sequential limits, By Heine Theorem,
  • Local Sign-Preserving Property: If and , then in a certain deleted neighborhood of , there always exists .
Pf. Proof by contradiction: Assume , s.t. Take , s.t. Take , s.t. Obtain a sequence satisfying and By Heine Theorem and the sign-preserving property of sequential limits, This contradicts .

Corollary:

  • If , then , s.t.
  • If , and , s.t. , , then .
  • If , then , s.t. is bounded in .

3.3 Single-Sided Limit

Definition: Let . If , it holds . Then is called the left-hand limit of at . Denoted as or .

Theorem. exists if and only if exist and are equal.

e.g. Heaviside Function

Functional Limit Definitions can be expanded. Consider a limit

“A” can be
“B” can be

e.g. .

3.4 Continuous Functions

3.5 Infinite Quantities

3.6 Continuous Function on Close Interval

References:

[1] 陈纪修, 数学分析, 第三版. 北京: 高等教育出版社, 2019.
[2] В. А. Зорич, Математический анализ, части I, II, 7-е изд. Москва: МЦНМО, 2015. (中译本: 数学分析, 第一、二卷. 李植 译. 北京: 高等教育出版社, 2019.)

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Calculus
  1. 1. Set Theory and Functions
    1. 1.1 Generalized Distributive and De Morgan’s Laws
    2. 1.2 Fundamental Properties
  2. 2. Sequence
    1. 2.1 Continuity of Real Number
    2. 2.2 Sequence Limits and Infinitesimals
    3. 2.3 Arthematic Operations of Limit
    4. 2.4 Infinite Sequences
    5. 2.5 Convergence Criteria (Real Number Continuity Theorem)
    6. 2.7 Subsequence
    7. 2.8 Cauchy Convergence Principle
  3. 3. Functional Limit
    1. 3.1 Definition and Properties
    2. 3.2 Properties of Functional Limit
    3. 3.3 Single-Sided Limit
    4. 3.4 Continuous Functions
    5. 3.5 Infinite Quantities
    6. 3.6 Continuous Function on Close Interval
  4. References: